VITEEE 2017 :: Mathematics :: General questions

• Mathematics - General questions

A straight line parallel to the line 2x - y+ 5 : 0 is also a tangent to the curve y² = 4x+ 5. Then the point of contact is

Answer:

Explanation:

Given curve  is y² = 4x+5 on differentiating, we get

2y $\frac{dy}{dx}$ = 4

$\frac{dy}{dx} = \frac{2}{y}$

Given line is 2x - y + 5 = 0

Y = 2x + 5

slope of line is 2. Therefore,

2/y = 2 and y = 1

put y = 1 in the equation of curve, we get

1 = 4x +5

x = -1

Hence, point of contact is (- 1, 1)

The acceleration of a sphere falling through a liquid is(30-3v) cm/s² where v is its speed in cm/ s. The maximum possible velocity of the sphere and the time when it is achieved are

Answer:

Explanation:

The differential equation of the motion is

$\frac{dv}{dt} = 30-3v ...(1)$

$\frac{dv}{30-3v} = dt$

Integrating we get 

$-\frac{1}{3} \log_{}{}( 30-3v)=t+C$

log(30-3v) = -3(t +c)

30-3v = $e^{-3t-3c}$ = A$e^{-3t}$, A = $e^{-3c}$

3v = 30 - A$e^{-3t}$ ....(2)

For maximum velocity $\frac{dv}{dt} = 0$

30-3v = 0 from (1)

.'. v = 30/3 = 10 cm/s which is the maximum velocity

However from (2) $\frac{dv}{dt} $= 3A$e^{-3t}$

Clearly $\frac{dv}{dt} $ = 0 , if t →∞

The maximum velocity will be achieved after infinite time in other words, the maximum velocity will never be reached.

The parabola having its focus at (3, 2) and directrix along the y-axis has its vertex at

Answer:

Explanation:

Vertex of the parabola, is a point which lies on the axis of the parabola, which is a line perpendicular to the directrix through the focus, i.e., y =  2 and equidistant from the focus and directrix x = 0, so that the vertex is (3/2, 2)

Ten different letters of an alphabet are given, words with five letters are formed from these given letters. Then the number of words that have at least one letter repeated is

Answer:

Explanation:

The total number of words that can be formed = 10⁵.

The number of words in which no letter is repeated = ¹⁰P₅. So, the number of words in which at least one letter is repeated = 10⁵ - ¹⁰P₅  = 69760.

Equation $\frac{1}{r}=\frac{1}{8}+\frac{3}{8}\cos\theta$ represents

Answer:

Explanation:

Given, equation is

$\frac{1}{r}=\frac{1}{8}+\frac{3}{8}\cos\theta$ or 8/r = 1 + 3cosθ

which is the form of l/r = 1+ ecosθ

'.' e = 3>0, .'. Given equation is a hyperbola

• Mathematics - General questions